# Iterative process for a strictly pseudo-contractive mapping in uniformly convex Banach spaces

- Yu Zhou
^{1}Email author, - Haiyun Zhou
^{1, 2}and - Peiyuan Wang
^{1}

**2014**:377

https://doi.org/10.1186/1029-242X-2014-377

© Zhou et al.; licensee Springer. 2014

**Received: **14 February 2014

**Accepted: **7 August 2014

**Published: **29 September 2014

## Abstract

This paper is concerned with a new method to prove the weak convergence of a strictly pseudo-contractive mapping in a *p*-uniformly convex Banach space with more relaxed restrictions on the parameters. Our results extend and improve the corresponding earlier results.

**MSC:**41A65, 47H17, 47J20.

## Keywords

## 1 Introduction and preliminaries

In 1967, Browder and Petryshyn [1] gave the classical definition for strictly pseudo-contractive mappings in Hilbert spaces for the first time.

**Definition 1.1**Let

*C*be a nonempty closed convex subset of a real Hilbert space

*H*. $T:C\to H$ is called a Browder-Petryshyn-type

*k*-strictly pseudo-contractive mapping. Then there exists $k\in [0,1)$ such that for every $x,y\in C$

In 2010, Zhou [2] gave a new definition for *k*-strictly pseudo-contractive mappings in *q*-uniformly smooth Banach spaces.

**Definition 1.2**Let

*C*be a nonempty closed convex subset of a

*q*-uniformly smooth Banach space

*X*. $T:C\to C$ is called a Zhou-type

*k*-strictly pseudo-contractive mapping, if there exists $k\in [0,1)$ such that for every $x,y\in C$

In 2009, Hu and Wang [3] gave another definition for *k*-strictly pseudo-contractive mappings in *p*-uniformly convex Banach spaces.

**Definition 1.3**Let

*C*be a nonempty closed convex subset of a

*p*-uniformly convex Banach space

*X*. $T:C\to C$ is called a Hu-type

*k*-strictly pseudo-contractive mapping, if there exists $k\in [0,1)$ such that for every $x,y\in C$

**Remark 1.1** The mappings defined by (1.1) and (1.2) are pseudo-contractive mappings, but the mapping defined by (1.3) may not be pseudo-contractive in general Banach spaces.

**Remark 1.2** If and only if $q=2$, the mappings defined by (1.1) and (1.2) are equivalent.

**Remark 1.3** If $p=q=2$, the mappings defined by (1.1), (1.2), and (1.3) are equivalent in Hilbert space.

In 1979, Reich [4] established a weak convergence theorem via a Mann-type iterative process for nonexpansive mapping in a uniformly convex Banach space with Fréchet differentiable norm.

**Theorem R** *Let* *C* *be a closed convex subset of a uniformly convex Banach space* *X* *with a Fréchet differentiable norm and* $T:C\to C$ *a nonexpansive mapping with* $F(T)\ne \mathrm{\varnothing}$. *For any* ${x}_{1}\in C$, *the iterative sequence* $\{{x}_{n}\}$ *is defined by* ${x}_{n+1}=(1-{\alpha}_{n}){x}_{n}+{\alpha}_{n}T{x}_{n}$, *where the real sequence* $\{{\alpha}_{n}\}\subset [0,1]$ *and* ${\sum}_{n=1}^{\mathrm{\infty}}(1-{\alpha}_{n}){\alpha}_{n}=\mathrm{\infty}$. *Then the sequence* $\{{x}_{n}\}$ *converges weakly to a fixed point of* *T*.

In 2007, Marino and Xu [5] improved Reich’s [4] result and gave several weak convergence theorems via the normal Mann iterative algorithm for strictly pseudo-contractive mappings in Hilbert spaces. Further, they proposed an open problem: *Do the main results of* [5]*still hold true in the framework of Banach spaces which are uniformly convex and have a Fréchet differentiable norm?*

In 2009, Hu and Wang [3] considered above problem in a *p*-uniformly convex Banach space and established the following theorem.

**Theorem H** *Let* *C* *be a closed convex subset of a* *p*-*uniformly convex Banach space* *X* *with a Fréchet differentiable norm and* $T:C\to C$ *be a* *k*-*strictly pseudo*-*contractive mapping in the light of* (1.3) *with coefficients* $p,k<min\{1,{2}^{-(p-2)}{c}_{p}\}$ *and* $F(T)\ne \mathrm{\varnothing}$. *For any* ${x}_{1}\in C$ *and* $n>1$, *the iterative sequence* $\{{x}_{n}\}$ *is defined by* ${x}_{n+1}=(1-{\alpha}_{n}){x}_{n}+{\alpha}_{n}T{x}_{n}$, *where the real sequence* $\{{\alpha}_{n}\}\subset [0,1]$ *and* $0<\epsilon \le {\alpha}_{n}\le 1-\epsilon <1-\frac{{2}^{p-2}k}{{c}_{p}}$. *Then the sequence* $\{{x}_{n}\}$ *converges weakly to a fixed point of* *T*.

**Question** Can one relax the restriction on the parameters ${\alpha}_{n}$ in Theorem H and simplify its proof?

The purpose of this paper is to solve the question mentioned above. To prove our results, we need the following lemmas.

**Lemma 1.1** (see [3])

*Let* *C* *be a nonempty closed convex subset of a* *p*-*uniformly convex Banach space* *X* *and* $T:C\to C$ *be a Hu*-*type strictly pseudo*-*contractive mapping in the light of* (1.3). *For* $\alpha \in (0,1)$, *define* ${T}_{\alpha}:C\to C$ *by* ${T}_{\alpha}=(1-\alpha )x+\alpha Tx$, *for* $x\in C$. *If* $\alpha \in (0,1-(k{2}^{p-2})/{c}_{p})$, *then* ${T}_{\alpha}$ *is a nonexpansive mapping and* $F({T}_{\alpha})=F(T)$.

**Lemma 1.2** (see [3])

*Let*

*C*

*be a nonempty closed convex subset of a*

*p*-

*uniformly convex Banach space*

*X*

*and*$T:C\to C$

*be a Hu*-

*type strictly pseudo*-

*contractive mapping in the light of*(1.3).

*For*$\mu \in (0,1)$, ${T}_{\mu}:C\to C$

*is defined by*${T}_{\mu}=(1-\mu )x+\mu Tx$,

*for*$x\in C$.

*Then the following inequality holds*:

*where* ${W}_{p}(\mu )={\mu}^{p}(1-\mu )+\mu {(1-\mu )}^{p}$.

**Lemma 1.3** (see [6])

*Let* *C* *be a nonempty closed convex subset of a* *p*-*uniformly convex Banach space* *X* *and* $T:C\to C$ *be a nonexpansive mapping*, *then* $I-T$ *is demiclosed at zero*.

**Lemma 1.4** (see [7])

*Let*

*C*

*be a nonempty closed convex subset of a*

*p*-

*uniformly convex Banach space*

*X*

*which satisfies the Opial condition and*$T:C\to C$

*be a quasi*-

*nonexpansive mapping with*$F(T)\ne \mathrm{\varnothing}$.

*If*$I-T$

*is demiclosed at zero*,

*then for any*${x}_{0}\in C$,

*the normal Mann iteration*$\{{x}_{n}\}$

*defined by*

*converges weakly to a fixed point of* *T*, *where* $\{{\alpha}_{n}\}\subset [0,1]$ *and* ${\sum}_{n=0}^{\mathrm{\infty}}min\{{\alpha}_{n},(1-{\alpha}_{n})\}=\mathrm{\infty}$.

**Lemma 1.5** (see [7])

*Let*

*C*

*be a nonempty closed convex subset of a*

*p*-

*uniformly convex Banach space*

*X*

*whose dual space*${X}^{\ast}$

*satisfies Kadec*-

*Klee property and*$T:C\to C$

*be a nonexpansive mapping with*$F(T)\ne \mathrm{\varnothing}$.

*Then*,

*for any*${x}_{0}\in C$,

*the normal Mann iteration*$\{{x}_{n}\}$

*defined by*

*converges weakly to a fixed point of* *T*, *where* $\{{\alpha}_{n}\}\subset [0,1]$ *and* ${\sum}_{n=0}^{\mathrm{\infty}}min\{{\alpha}_{n},(1-{\alpha}_{n})\}=\mathrm{\infty}$.

Now we are in a position to state and prove the main results in this paper.

## 2 Main results

**Theorem 2.1**

*Let*

*C*

*be a nonempty closed convex subset of a*

*p*-

*uniformly convex Banach space*

*X*

*with Fréchet differential norm*.

*Let*$T:C\to C$

*be a Hu*-

*type*

*k*-

*strictly pseudo*-

*contractive mapping in the light of*(1.3)

*with coefficients*$p,k<min\{1,{2}^{-(p-2)}{c}_{n}\}$

*and*$F(T)\ne \mathrm{\varnothing}$.

*Assume that a real sequence*$\{{\alpha}_{n}\}$

*in*$[0,1]$

*satisfies the conditions*:

- (i)
$0\le {\alpha}_{n}\le \alpha =1-(k{2}^{p-2}/{c}_{p})$, $n\ge 0$;

- (ii)
${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}[(1-{\alpha}_{n}){2}^{2-p}{c}_{p}-k]=\mathrm{\infty}$.

*For any*${x}_{0}\in C$,

*the normal Mann iterative sequence*$\{{x}_{n}\}$

*is defined by*

*Then the sequence* $\{{x}_{n}\}$ *defined by* (2.1) *converges weakly to a fixed point of* *T*.

*Proof* Let ${T}_{\alpha}$ be given as in Lemma 1.1. Then ${T}_{\alpha}:C\to C$ is a nonexpansive mapping with $F({T}_{\alpha})=F(T)$. Set ${\beta}_{n}=\frac{\alpha -{\alpha}_{n}}{\alpha}$. Then (2.1) reduces to ${x}_{n+1}={\beta}_{n}{x}_{n}+(1-{\beta}_{n}){T}_{\alpha}{x}_{n}$.

By using Theorem R, we conclude that $\{{x}_{n}\}$ converges weakly to a fixed point of ${T}_{\alpha}$, and of *T*. The proof is complete. □

**Remark 2.2** Theorem 2.1 relaxes the iterative parameters in Theorem H and our proof method is also quite concise.

**Theorem 2.3**

*Let*

*C*

*be a nonempty closed convex subset of a*

*p*-

*uniformly convex Banach space*

*X*

*which satisfies the Opial condition*.

*Let*$T:C\to C$

*be a Hu*-

*type*

*k*-

*strictly pseudo*-

*contractive mapping in the light of*(1.3)

*with coefficients*$p,k<min\{1,{2}^{-(p-2)}{c}_{p}\}$

*and*$F(T)\ne \mathrm{\varnothing}$.

*Assume that the real sequence*$\{{\alpha}_{n}\}$

*in*$[0,1]$

*satisfies the conditions*:

- (i)
$0\le {\alpha}_{n}\le \alpha =1-(k{2}^{p-2}/{c}_{p})$, $n\ge 0$;

- (ii)
${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}[(1-{\alpha}_{n}){2}^{2-p}{c}_{p}-k]=\mathrm{\infty}$.

*For any*${x}_{0}\in C$,

*the normal Mann iteration*$\{{x}_{n}\}$

*is defined by*

*Then the sequence* $\{{x}_{n}\}$ *defined by* (2.2) *converges weakly to the fixed point of* *T*.

*Proof* Let ${T}_{\alpha}$ be given as in Lemma 1.1. Then ${T}_{\alpha}:C\to C$ is a nonexpansive mapping with $F({T}_{\alpha})=F(T)$. Set ${\beta}_{n}=\frac{\alpha -{\alpha}_{n}}{\alpha}$. Then (2.2) reduces to ${x}_{n+1}={\beta}_{n}{x}_{n}+(1-{\beta}_{n}){T}_{\alpha}{x}_{n}$. As shown in Theorem 2.1, ${\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}(1-{\beta}_{n})=\mathrm{\infty}$. By Lemma 1.3, $I-{T}_{\alpha}$ is demiclosed at zero. By Lemma 1.4, we conclude that $\{{x}_{n}\}$ converges weakly to a fixed point of ${T}_{\alpha}$, and of *T*. The proof is complete. □

**Theorem 2.4**

*Let*

*C*

*be a nonempty closed convex subset of a*

*p*-

*uniformly convex Banach space*

*X*

*with the dual space*${X}^{\ast}$

*satisfying the Kadec*-

*Klee property*.

*Let*$T:C\to C$

*be a Hu*-

*type*

*k*-

*strictly pseudo*-

*contractive mapping in the light of*(1.3)

*with coefficients*$p,k<min\{1,{2}^{-(p-2)}{c}_{p}\}$

*and*$F(T)\ne \mathrm{\varnothing}$.

*Assume that the real sequence*$\{{\alpha}_{n}\}$

*in*$[0,1]$

*satisfies the conditions*:

- (i)
$0\le {\alpha}_{n}\le \alpha =1-(k{2}^{p-2}/{c}_{p})$, $n\ge 0$;

- (ii)
${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}[(1-{\alpha}_{n}){2}^{2-p}{c}_{p}-k]=\mathrm{\infty}$.

*For any*${x}_{0}\in C$,

*the normal Mann iteration*$\{{x}_{n}\}$

*is defined by*

*Then the sequence* $\{{x}_{n}\}$ *defined by* (2.3) *converges weakly to a fixed point of* *T*.

*Proof* Let ${T}_{\alpha}$ be given as in Lemma 1.1. Then ${T}_{\alpha}:C\to C$ is a nonexpansive mapping with $F({T}_{\alpha})=F(T)$. Set ${\beta}_{n}=\frac{\alpha -{\alpha}_{n}}{\alpha}$. Then (2.3) reduces to ${x}_{n+1}={\beta}_{n}{x}_{n}+(1-{\beta}_{n}){T}_{\alpha}{x}_{n}$. As shown in Theorem 2.1, ${\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}(1-{\beta}_{n})=\mathrm{\infty}$. By using Lemma 1.5, $\{{x}_{n}\}$ defined by (2.3) converges weakly to a fixed point of ${T}_{\alpha}$, and of *T*. The proof is complete. □

## Declarations

### Acknowledgements

This research was supported by the National Natural Science Foundation of China (11071053).

## Authors’ Affiliations

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